Stetson.edu informs us that 778 is the number of ways a 1 x 5 rectangle can be surrounded by other 1 x 5 rectangles. I’m not going to publish or even find all 778 possible ways, but here is one of them:

27² + 7² = 778.

We can use those numbers to compute a Pythagorean triple: 2(27)(7), 27² – 7², 27² + 7², which turns out to be 378-680-778, and thus 378² + 680² = 778².